# A Mixed Logical Dynamical-Model Predictive Control (MLD-MPC) Energy Management Control Strategy for Plug-in Hybrid Electric Vehicles (PHEVs)

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Modeling of Powertrain System

^{3}), ${r}_{\mathrm{w}}$ is the radius of wheel in m, ${\mathsf{\omega}}_{w}$ is the wheel rotating speed in rad/s, ${T}_{W}$ is the wheel torque in $\mathrm{N}\cdot \mathrm{m}$, ${A}_{f}$ is the frontal area in ${\mathrm{m}}^{2}$, ${C}_{d}$ is the drag coefficient, ${f}_{r}$ represents tire rolling resistance coefficient, m is the vehicle mass in kg, $J$ represents total vehicle rotational inertia in $\mathrm{kg}\cdot {\mathrm{m}}^{2}$, $\mathsf{\alpha}$ is vehicle slope angle in rad, $\mathrm{d}v/\mathrm{d}t$ is the acceleration in m/s

^{2}. Model of transmission torque and speed is given in Equation (2), where ${\omega}_{in}$ is the required input rotating speed of transmission in rad/s; ${i}_{0}$ is the total transmission ratio; ${T}_{in}$ is the required input torque of transmission in $\mathrm{N}\cdot \mathrm{m}$; $\mathsf{\eta}$ is the efficiency from transmission input to tire [26]. The model of engine fuel consumption rate is given in Equation (3), where ${\dot{m}}_{f}$ is the engine fuel consumption rate, which is a function of its rotation speed ${\mathsf{\omega}}_{e}$ and torque ${T}_{e}$:

## 3. Short-Term Vehicle Speed Prediction

#### 3.1. Driving Intention Identification

#### 3.2. Short-Term Vehicle Speed Prediction Using Nonlinear Auto-Regressive Neural Network

## 4. Mixed Logic Dynamical Model Predictive Control Strategy

#### 4.1. Modeling of Mixed Logic Dynamical Predictive Control Strategy

- (1)
- at time k, predict the short-term future speed profile for the current control horizon (k–k + N, where N is the receding horizon) and calculate the corresponding required torque through Equations (1) and (2);
- (2)
- the MLD model calculates the optimal control policy (u(k)–u(k + N)) for the current prediction horizon (k–k + N);
- (3)
- apply the first time-step of the optimal control policy u(k) in the controlled PHEV model;
- (4)
- update the state variable and system constraints, repeat the computation procedure 1–3 at the next time instant (time k + 1).

- $\begin{array}{cc}\hfill {B}_{1}& =[{b}_{0}(k)\text{\hspace{1em}}0\text{\hspace{1em}}{b}_{0}(k)\text{\hspace{1em}}{b}_{0}(k)\text{\hspace{1em}}{b}_{0}(k)\text{\hspace{1em}}0]\hfill \end{array}$
- $\begin{array}{cc}\hfill {B}_{2}& =[{b}_{1}(k)\text{\hspace{1em}}0\text{\hspace{1em}}{b}_{1}(k)\text{\hspace{1em}}{b}_{1}(k)\text{\hspace{1em}}{b}_{1}(k)\text{\hspace{1em}}0]\hfill \end{array}$
- $\begin{array}{cc}\hfill {D}_{1}& =[-\mathsf{\lambda}{U}_{0}{b}_{0}(k){Q}_{\mathrm{max}}/R,\text{\hspace{0.17em}}{a}_{0}(k)+{a}_{1}(k){T}_{in}(k),\text{\hspace{0.17em}}-\mathsf{\lambda}{U}_{0}{b}_{0}(k){Q}_{\mathrm{max}}/R+{a}_{0}(k)+{a}_{1}(k){T}_{in}(k),\hfill \\ & \text{\hspace{1em}}-\mathsf{\lambda}{U}_{0}{b}_{0}(k){Q}_{\mathrm{max}}/R+{a}_{0}(k)+{a}_{1}(k){T}_{in}(k),\text{\hspace{0.17em}}0,\text{\hspace{0.17em}}0]\hfill \end{array}$
- $\begin{array}{cc}\hfill {D}_{2}& =[-\mathsf{\lambda}{U}_{1}{b}_{0}(k){Q}_{\mathrm{max}}/R,\text{\hspace{0.17em}\hspace{0.17em}}0,\text{\hspace{0.17em}}-\mathsf{\lambda}{U}_{0}{b}_{1}(k){Q}_{\mathrm{max}}/R-{i}_{t}{a}_{1}(k),\hfill \\ & \text{\hspace{1em}}-\mathsf{\lambda}{U}_{0}{b}_{1}(k){Q}_{\mathrm{max}}/R-{i}_{t}{a}_{1}(k)-\mathsf{\lambda}{U}_{0}{b}_{1}(k){Q}_{\mathrm{max}}/R,0]\hfill \end{array}$

**B**

_{1},

**B**

_{2}is used to calculate the SOC change corresponding to the specific motor torque, b

_{0}(k) and b

_{1}(k) are the fitting coefficients for the specific motor speed at time k, and they are obtained from Equation (8). Similarly,

**D**

_{1}and

**D**

_{2}is used to calculate the sum of engine fuel consumption and motor equivalent fuel consumption, a

_{0}(k) and a

_{1}(k) are from Equation (4):

**E**

_{1},

**E**

_{2},

**E**

_{3}and

**E**

_{4}are all the coefficients in the form of Matrix converted from Table 4.

**B**

_{1},

**B**

_{2},

**D**

_{1},

**D**

_{2}is the same as Equation (10), and

**E**

_{1},

**E**

_{2},

**E**

_{3},

**E**

_{4}, is defined as follows:

**B**

_{1},

**B**

_{2},

**D**

_{1},

**D**

_{2},

**E**

_{1},

**E**

_{2},

**E**

_{3}and

**E**

_{4}at every sampling time k and according to the objective function of minimal equivalent fuel consumption, Equations (13) and (14) can be converted into a MILP problem.

#### 4.2. Solution of Mixed Integer Linear Programming

## 5. Simulation Experiments and Analysis

#### 5.1. Simulation Experiments

_{m}is 272 N·m, and M

_{in}is 1170 N·m. Fitting coefficients of engine and motor are listed in the Appendix A. ADVISOR (Advanced Vehicle Simulator, 2002, National Renewable Energy Laboratory, Golden, CO, USA) is used to simulate the UKBUS driving cycle as shown in Figure 11, with an initial SOC value of 0.7 ($0.3\le SOC\le 0.8$), prediction horizon N of 15 s and sampling period of 1 s (the effect of predict time domain is discussed in Section 5.2). The simulation results are presented in Figure 12. Figure 12a is the comparison of the SOC curves between the MLD-MPC control strategy proposed in this paper and Charge Depleting-Charge Sustaining (CD-CS) control strategy. As can be observed, the CD-CS strategy consumes the battery energy within 8000 s, and then sustains SOC around 0.3 for the remainder of the trip. In the MLD-MPC control strategy, the battery energy is completely consumed by the end of the trip. Compared to CD-CS control strategy, the proposed MLD-MPC strategy can make more reasonable use of battery energy. Figure 12b is the comparison of the fuel consumption between the MLD-MPC control strategy presented in this paper and CD-CS control strategy. As seen in the picture, the CD-CS control strategy makes full use of battery energy and does not consume fuel within the first 8000 s, then the fuel consumption grow rapidly during the Charge-Sustaining mode, while the MLD-MPC control strategy maintains a relatively slow fuel consumption rate. Up to the end of the whole driving cycle, the fuel consumption is 12.93 L, and the end value of SOC is 0.3043 for the MLD-MPC control strategy, while the fuel consumption is 15.39 L, and the end value of SOC is 0.3125 for the CD-CS control strategy. It can be concluded from this results that under the same driving cycle conditions, the MLD-MPC control strategy reduces the fuel consumption of a PHEV. Figure 13 shows the comparison of engine operation points of these two control strategies. As can be observed, the engine operation points of the CD-CS control strategy spread out on a much larger area, and a lot of operation points in the low efficiency area. While the engine operation points of MLD-MPC control strategy is relatively centralized and most of them are in the high efficiency area which will improve fuel economic.

#### 5.2. Influence of Prediction Horizon for Mixed Logic Dynamical-Model Predictive Control (MLD-MPC) Strategy

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

- a
_{1}= [0.0037, 0.0045, 0.0050, 0.0055, 0.0060, 0.0065, 0.0069, 0.0073, 0.0078, 0.0083, 0.0088, 0.0094, 0.0100, 0.0107, 0.0113, 0.0122, 0.0130]; - a
_{0}= [0.1021, 0.1232, 0.1372, 0.1513, 0.1654, 0.1795, 0.2258, 0.2771, 0.3053, 0.3347, 0.3685, 0.4039, 0.4412, 0.4800, 0.5453, 0.5856, 0.6275]

M_V = | [1000 2000 3000 4000 5000 6000 7000 8000 9000 10000]; |

[–1000 –2000 –3000 –4000 –5000 –6000 –7000 –8000 –9000 –10000] |

b_{1} | = | 1.0 × 10^{−5} × [−0.0406, −0.0751, −0.1120, −0.1591, −0.2056, −0.2582, −0.2989, −0.3413, −0.3779, −0.4234]; |

1.0 × 10^{−5} × [−0.0246, −0.0535, −0.0805, −0.1002, −0.1222, −0.1396, −0.1626, −0.1862, −0.2117, −0.2355]; | ||

b_{0} | = | 1.0 × 10^{−4} × [0.0046, 0.0002, 0.0132, 0.0604, 0.0824, 0.1037, 0.0777, 0.0600, 0.0435, 0.0343]; |

1.0 × 10^{−4} × [0.0074, 0.0047, 0.0255, 0.0857, 0.1048, 0.1310, 0.1194, 0.1005, 0.0888, 0.0668] |

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**Figure 1.**Parallel plug-in hybrid electric vehicle (PHEV) powertrain and transmission system (1—Touque Coupler; 2—Clutch; 3—Engine; 4—Motor; 5—Energy Storage System; 6—Gearbox; 7—Final Drive; 8—Wheels).

**Figure 4.**(

**a**) Membership functions of acceleration pedal travel; (

**b**) membership functions of rate of change of acceleration pedal travel; and (

**c**) membership functions of acceleration intention.

**Figure 5.**(

**a**) Membership functions of braking pedal travel; (

**b**) membership functions of rate of change of braking pedal travel; and (

**c**) membership functions of braking intention.

**Figure 12.**(

**a**) comparison of SOC curves of two strategies; and (

**b**) comparison of the fuel consumption of two strategies.

**Figure 13.**(

**a**) Engine operation points of charge depleting-charge sustaining (CD-CS) strategy; and (

**b**) engine operation points of mixed logic dynamical-model predictive control (MLD-MPC) Strategy.

Pedal Travel | Rate of Change of Acceleration Pedal | ||||
---|---|---|---|---|---|

Negative Big | Negative Small | Small | Middle | Big | |

Small | slow | slow | relatively slow | relatively slow | normal |

Relatively Small | slow | relatively slow | relatively slow | normal | relatively rapid |

Middle | relatively slow | normal | normal | relatively rapid | rapid |

Relatively Big | relatively slow | normal | relatively rapid | relatively rapid | rapid |

Big | normal | normal | relatively rapid | rapid | rapid |

Pedal Travel | Rate of change of Braking Pedal | ||||
---|---|---|---|---|---|

Negative Big | Negative Small | Small | Middle | Big | |

Small | slow | slow | relatively slow | relatively slow | normal |

Relatively Small | slow | relatively slow | relatively slow | normal | normal |

Middle | relatively slow | relatively slow | normal | normal | relatively rapid |

Relatively Big | relatively slow | normal | relatively rapid | relatively rapid | rapid |

Big | normal | normal | relatively rapid | rapid | rapid |

Model Input Type | Prediction Horizon and Result Type | Prediction Results | |||||||
---|---|---|---|---|---|---|---|---|---|

NAR Model Using Driving Intention Data | Prediction Horizon (s) | 1 | 2 | 5 | 10 | 20 | 30 | 40 | 50 |

RMSE | 0.6416 | 1.1260 | 2.8917 | 5.6310 | 9.0878 | 10.9024 | 12.0005 | 12.6537 | |

NAR Model not Using Driving Intention Data | Prediction Horizon (s) | 1 | 2 | 5 | 10 | 20 | 30 | 40 | 50 |

RMSE | 0.8757 | 1.4081 | 3.1606 | 6.0167 | 10.0957 | 12.6161 | 13.9765 | 14.5821 |

Electric Power Mode | Drive Charging Mode |

$\{\begin{array}{c}{M}_{in}{\mathsf{\delta}}_{1}\le {M}_{in}+{T}_{in}-\mathsf{\epsilon}\hfill \\ {M}_{in}{\mathsf{\delta}}_{1}\le {M}_{in}-{T}_{in}+{i}_{t}{T}_{m\_\mathrm{max}}\hfill \\ -{T}_{in}{\mathsf{\delta}}_{1}+{i}_{t}{z}_{1}\le 0\hfill \\ {T}_{in}{\mathsf{\delta}}_{1}-{i}_{t}{z}_{1}\le 0\hfill \end{array}$ | $\{\begin{array}{c}{M}_{in}{\mathsf{\delta}}_{4}\le -{i}_{t}u+{M}_{in}+{T}_{in}-\mathsf{\epsilon}\hfill \\ {M}_{in}{\mathsf{\delta}}_{4}\le {i}_{t}u+{M}_{in}-{T}_{in}+{T}_{e\_\mathrm{max}}\hfill \\ {M}_{m}{\mathsf{\delta}}_{4}\le u+{M}_{m}+{T}_{m\_\mathrm{max}}\hfill \\ {M}_{m}{\mathsf{\delta}}_{4}\le -u+{M}_{m}-\mathsf{\epsilon}\hfill \end{array}$ |

Fuel Consumption Mode | Regenerate Braking Mode |

$\{\begin{array}{c}{M}_{in}{\mathsf{\delta}}_{2}\le {M}_{in}+{T}_{in}-\mathsf{\epsilon}\hfill \\ {M}_{in}{\mathsf{\delta}}_{2}\le {M}_{in}-{T}_{in}+{T}_{e\_\mathrm{max}}\hfill \\ {z}_{2}\le 0\hfill \\ -{z}_{2}\le 0\hfill \end{array}$ | $\{\begin{array}{c}{M}_{in}{\mathsf{\delta}}_{5}\le {M}_{in}-{T}_{in}-\mathsf{\epsilon}\hfill \\ -{T}_{acc\_\mathrm{max}}{\mathsf{\delta}}_{5}+{z}_{5}\le 0\hfill \\ {T}_{acc\_\mathrm{max}}{\mathsf{\delta}}_{5}-{z}_{5}\le 0\hfill \end{array}$ |

Mixed Driven Mode | Stopping Mode |

$\{\begin{array}{c}{M}_{in}{\mathsf{\delta}}_{3}\le -{i}_{t}u+{M}_{in}+{T}_{in}-\mathsf{\epsilon}\hfill \\ {M}_{in}{\mathsf{\delta}}_{3}\le {i}_{t}u+{M}_{in}-{T}_{in}+{T}_{e\_\mathrm{max}}\hfill \\ {M}_{m}{\mathsf{\delta}}_{3}\le u+{M}_{m}-\mathsf{\epsilon}\hfill \\ {M}_{m}{\mathsf{\delta}}_{3}\le -u+{M}_{m}+{T}_{m\_\mathrm{max}}\hfill \end{array}$ | $\{\begin{array}{c}{T}_{in}{\mathsf{\delta}}_{6}\le 0\hfill \\ -{T}_{in}{\mathsf{\delta}}_{6}\le 0\hfill \\ {z}_{6}\le 0\hfill \\ -{z}_{6}\le 0\hfill \end{array}$ |

Driving Condition | Operation Mode | Motor Torque Decision |
---|---|---|

${T}_{in}=0$ | ${\mathsf{\delta}}_{6}=1$ | $u=0$ |

${T}_{in}<0$ | ${\mathsf{\delta}}_{5}=1$ | $u={T}_{in}/{i}_{t}$ |

$SOC<SO{C}_{\mathrm{min}}$ | ${\mathsf{\delta}}_{4}=1$ | $u=({T}_{in}-{T}_{e\mathrm{max}})/{i}_{t}$ |

${T}_{in}>{T}_{e\_\mathrm{max}}$ | ${\mathsf{\delta}}_{3}=1$ | $u>0,\text{\hspace{0.17em}}0<{T}_{in}-{i}_{t}u<{T}_{e\_\mathrm{max}}$ |

Else | ${\mathsf{\delta}}_{1}=1$ | $u={T}_{in}/{i}_{t}$ |

${\mathsf{\delta}}_{2}=1$ | $u=0$ | |

${\mathsf{\delta}}_{3}=1$ | $u>0,\text{\hspace{0.17em}}0<{T}_{in}-{i}_{t}u<{T}_{e\_\mathrm{max}}$ | |

${\mathsf{\delta}}_{4}=1$ | $u<0,0<{T}_{in}-{i}_{t}u<{T}_{e\_\mathrm{max}}$ |

Parameter | Value | Unit |
---|---|---|

Maximum mass | 13,485 | kg |

Air resistance coefficient | 0.79 | - |

Frontal area | 7.24 | m^{2} |

Wheel rolling radius | 0.5 | m |

Rolling resistance coefficient | 0.0094 | - |

Wheel base | 6.85 | m |

Final ratio | 1 | - |

Rotational inertia of wheels in total | 20.5215 | $\mathrm{kg}\cdot {\mathrm{m}}^{2}$ |

Rotational inertia of engine and motor | 2.2511 | $\mathrm{kg}\cdot {\mathrm{m}}^{2}$ |

**Table 7.**Fuel consumption under two control strategies. NEDC: New European Drive Cycle; OCC: Orange County Bus Cycle.

Driving Cycle | Control Strategy | Final SOC | Fuel Consumption (L) | Fuel Saved |
---|---|---|---|---|

5 × UKBUS | MLD-MPC | 0.3043 | 12.93 | 15.98% |

CD-CS | 0.3125 | 15.39 | ||

8 × NEDC | MLD-MPC | 0.2939 | 12.46 | 14.36% |

CD-CS | 0.2743 | 14.55 | ||

8 × OCC | MLD-MPC | 0.2959 | 15.95 | 7.27% |

CD-CS | 0.3023 | 17.20 |

Bottoming Time | First Bottoming | Second Bottoming | |
---|---|---|---|

Prediction Horizon | |||

1 s | 643 s | 847 s | |

5 s | 643 s | 849 s | |

10 s | 650 s | 851 s | |

15 s | 650 s | 976 s | |

20 s | 651 s | 976 s | |

25 s | 651 s | 976 s | |

30 s | 675 s | 1040 s |

© 2017 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lian, J.; Liu, S.; Li, L.; Liu, X.; Zhou, Y.; Yang, F.; Yuan, L.
A Mixed Logical Dynamical-Model Predictive Control (MLD-MPC) Energy Management Control Strategy for Plug-in Hybrid Electric Vehicles (PHEVs). *Energies* **2017**, *10*, 74.
https://doi.org/10.3390/en10010074

**AMA Style**

Lian J, Liu S, Li L, Liu X, Zhou Y, Yang F, Yuan L.
A Mixed Logical Dynamical-Model Predictive Control (MLD-MPC) Energy Management Control Strategy for Plug-in Hybrid Electric Vehicles (PHEVs). *Energies*. 2017; 10(1):74.
https://doi.org/10.3390/en10010074

**Chicago/Turabian Style**

Lian, Jing, Shuang Liu, Linhui Li, Xuanzuo Liu, Yafu Zhou, Fan Yang, and Lushan Yuan.
2017. "A Mixed Logical Dynamical-Model Predictive Control (MLD-MPC) Energy Management Control Strategy for Plug-in Hybrid Electric Vehicles (PHEVs)" *Energies* 10, no. 1: 74.
https://doi.org/10.3390/en10010074